Ramyak Bilas | Searcharxiv

arXiv2026

The Dennis Trace for Assembler K-Theory

We define a notion of Hochschild homology of a weak assembler and use it to construct a Dennis trace map from the group completion $K$-theory of a weak assembler to its Hochschild homology. We compute the Hochschild homology for some examples, in particular for various cases of restricted polytopes. We also give an explicit description of the Dennis trace map at level $0$, i.e. from the zeroth $K$-theory group to the zeroth Hochschild homology group. Further, we show that the Dennis trace is a refinement of the trace/regulator map defined by Bohmann et al thus showing that group homology is a trace invariant.

math.AT↗

arXiv2026

Beyond Linear and Overcomplete Regimes: A Mean-Field Analysis of Bottleneck Autoencoders

Autoencoders (AEs) learn low-dimensional representations by mapping data into a latent space while minimizing reconstruction error. Despite their empirical success, theoretical understanding remains limited and largely restricted to linear models or settings without a bottleneck. In this work, we study nonlinear AEs with a fixed finite-dimensional bottleneck in the mean-field (MF) regime. We derive explicit MF learning dynamics for both encoder and decoder, providing a tractable characterization of training in the nonlinear setting. We show that, over finite time horizons, the empirical risk of finite-width networks trained with stochastic gradient descent closely tracks the MF risk trajectory with high probability. At optimality, we further establish that the finite-width risk converges to the MF optimum, demonstrating that finite networks are sufficiently expressive to approximate the infinite-width solution.

cs.LG↗